The Momentum Kernel of Gauge and Gravity Theories

We derive an explicit formula for factorizing an $n$-point closed string amplitude into open string amplitudes. Our results are phrased in terms of a momentum kernel which in the limit of infinite string tension reduces to the corresponding field theory kernel. The same momentum kernel encodes the monodromy relations which lead to the minimal basis of color-ordered amplitudes in Yang-Mills theory. There are interesting consequences of the momentum kernel pertaining to soft limits of amplitudes. We also comment on surprising links between gravity and certain combinations of kinematic and color factors in gauge theory.

been provided. In this paper we will derive the explicit formulae for any n. A central object that emerges is a momentum kernel S α ′ [i 1 , . . . , i k |j 1 , . . . , j k ] p ≡ (πα ′ /2) −k k t=1 sin πα ′ (p · k it + k q>t θ(i t , i q ) k it · k iq ) , (1.1) whose precise definition will be provided below. This momentum kernel maps products of open string amplitudes to closed string amplitudes. In the field theory limit α ′ → 0 it turns into the field theory momentum kernel S that maps field theory Yang-Mills amplitudes to gravity amplitudes [2,3]. In beautiful agreement with the corresponding observation in field theory, the string theory momentum kernel S α ′ is precisely the generator of monodromy relations. Phrased more precisely, it annihilates color-ordered amplitudes A n according to σ S α ′ [σ(2, . . . , n − 1)|β(2, . . . , n − 1)] k 1 A n (n, σ(2, . . . , n − 1), 1) = 0 , (1.2) where β is any permutation of leg 2, . . . , n − 1 and the sum runs over all permutations of these legs. Particularly interesting insight arises if one uses a construction based on the heterotic string [1]. This shows that Yang-Mills amplitudes A Y M n can be expressed as a sum of products of color-ordered Yang-Mills amplitudes A n , A Y M n ∼ A n S α ′ A s n , where A s n is a scalar amplitude based on vertices that are trivial except for the structure constants of the gauge group [12]. The similarity with the relation for gravity amplitudes M n ∼ A n S α ′ A n in terms of gauge amplitudes is striking. This leads naturally to an alternative viewpoint on KLT-relations that comes from exploring Jacobi-like identities [5] among numerator factors on the Yang-Mills side. Also this picture, and its generalization to extended Jacobi-like structures, has a natural string theory interpretation [13][14][15]. This becomes particularly transparent within the framework of the heterotic string [14] (see also [16,17]). There is hope that much of this tree-level structure [18,19] carries over to any number of loops [20]. This can be used to analyze the ultraviolet behaviour of N = 8 supergravity [21]. All of this is strong motivation for analyzing the numerator approach and the associated duality between kinematic and color structures in greater detail.
Finally, an interesting case arises when one applies the field theory momentum kernel S to amplitudes with mismatched external helicity legs. Then one obtains new non-linear relations among Yang-Mills amplitudes [22] that can be understood in the context of Rcharges [23][24][25][26]. This paper will be devoted to a detailed study of the string theory momentum kernel S α ′ . First we derive, for the first time, the explicit n-point relations between closed string and open string amplitudes, and thus demonstrate in detail how the momentum kernel arises in string theory. We next explore some of the properties of this momentum kernel, and in particular we explain how it acts as a generator of monodromy relations. It turns out that there is, just as in field theory [22], a great amount of freedom in writing down the explicit KLT-map, a freedom which is directly related to this monodromy. Common to all rewritings is, however, the same momentum kernel S α ′ . In the last part of the paper we make some observations regarding the momentum kernel S in the field theory limit and soft factorization of amplitudes in gauge theory and gravity. We also comment on the relation to the approach based on Jacobi-like structures among amplitudes.

The momentum kernel in string theory
In this section we will derive a general form of n-point closed string amplitudes in terms of products of color-ordered open string amplitudes. Like in the KLT paper [1], we proceed by explicit holomorphic factorization of a closed string amplitude.
The heterotic and open strings have different spectra that leads to different effective actions. In the field theory α ′ → 0 limit the heterotic and open string tree-level amplitudes reduce to various gauge or gravity amplitudes. Closed type II and heterotic string tree-level gravity amplitudes are different because of the different spectra in the two theories. N = 8 supergravity amplitudes are computed as the field theory limit of closed string type II amplitudes, and N = 4 supergravity amplitudes as the field theory limit of heterotic string amplitudes. For multi-graviton amplitudes the difference arises at higher order in α ′ and it does not affect the field theory limit.
When performing the holomorphic factorization of the amplitudes we will not have to specify the actual detailed form of the left and right moving string amplitudes. But we will pay attention to the monodromy properties of the amplitude when the positions of the external states move on the sphere. All the monodromy properties [8,15] arise from the contraction between the corresponding plane-wave factors. In this way universal relations emerge.

Gravity and gauge theory amplitudes
After fixing the three points z 1 = 0, z n−1 = 1 and z n = ∞, the n-point closed string amplitude takes the general form where f (z i ) and g(z i ) arise from the operator product expansion of the vertex operators. They are functions without branch cuts. The precise form of these functions depends on the external states. They can be gravitons or gauge fields, or, in the straightforward supersymmetric generalization, any part of the N = 4 supermultiplet. However, the exact form of these functions will not affect the general relations. For definiteness, let us from now on consider M n to be a gravity amplitude.
Following [27,28] we can write eq. (2.1) in terms of v 1 i and v 2 i , where z i = v 1 i + iv 2 i , and then make the following change of variables for v 2 where ǫ > 0 is some small number. Keeping only the terms linear in ǫ, i.e. using 3) and introducing the notation we can write eq. (2.1) as an 'almost factorized' amplitude The appearance of branch cuts in the integrand is crucial. We have the following choices for x α when x < 0 By splitting the v + i -integrals and using symmetry, the n-point graviton amplitude can be written as where M n (σ(2), · · · , σ(n − 2)) is the ordered amplitude defined such that v + σ(2) < v + σ(3) < · · · < v + σ(n−2) . If one of the v + i is in the interval ] − ∞, 0[ the integral contours for v − i lie below the real axis and the integral vanishes. This is because there are no poles of the functions f and g outside the real axis, and the integrand nicely vanishes at infinity. By the same reasoning, if one of the v + i is in the interval ]1, +∞[, the integral contours for v − i lie above the real axis and the integral is again vanishing.
Therefore, to get a non-vanishing contribution, all the v + 's need to be distributed in the interval ]0, 1[: which are integrals corresponding to color-ordered open string amplitudes. We now turn to the evaluation of the integrals over the v − i variables. This will lead to ordered integrals A n that correspond to the 'right-moving' sectors. For simplicity we just consider the case of ordering {2, 3, . . . , n − 2} of the v + i variables. All other cases are obtained by summing over the permutations of these n − 3 variables.
For each 2 ≤ i ≤ n − 2 we first examine the behavior of the integrand of (2.1) around the different branch cuts.
i has a positive imaginary part. Therefore the contour is above the real axis.
has a negative imaginary part; the contour of integration lies below the real axis. Finally, has a negative imaginary part. Therefore the contour of integration for v − i goes below the contour of v − j for i < j. We have represented this nested structure of the contours of integration for the v − i variables in figure 1.
We now consider the deformations of the contours of integration for the v − i variables. Because the contours cannot cross each other we need to close them either to the right, turning around the branch cut at z = 1 by starting with the rightmost, or close the contours to the left, turning around the branch cut at z = 0, starting with the leftmost.
There is evidently an arbitrariness in the number of contours that are closed to the left or closed to the right. For a given 2 ≤ j ≤ n − 2, we can pull the contours for the set between 2 and j − 1 to the left, and the set between j and n − 2 to the right.
Pulling the contour for 2 gives (2.9) Here we have explicitly shown only the contributions where v − 2 has branch cuts.
Closing the contour for v − 3 to the left leads to and so on until one has pulled the contour for v − j−1 to the left. When closing the contours to the right we start from the contour for v − n−2 down to the one for v − j . Pulling first the contour for v − n−2 to the right leads to Similarly, closing the contour for v − n−3 to the right leads to and so on until one reaches the contour for v − j . The integrals over the v − variables are the ordered string amplitudes A n (γ(2, . . . , j − 1), 1, n − 1, β(j, . . . , n − 2), n).
Collecting these contour deformations lead to the following expression for the v − part of the integral in (2.1) which is of course multiplied by the left-moving amplitude A n (1, 2, . . . , n) from the integral over the v + variables. Here we also see the first appearance of the momentum kernel S α ′ [γ, σ] p . It depends on the permutations γ of the v − i -variables and the ordering of the v + i . It also depends on the momenta p = k 1 and k n−1 of the states at the branch cut at z = 0 or z = 1 on the sphere. In general it can be defined as where θ(i t , i q ) equals 1 if the ordering of the legs i t and i q is opposite in the sets {i 1 , . . . , i k } and {j 1 , . . . , j k }, and 0 if the ordering is the same.
We have normalized this expression so that in the field theory limit α ′ → 0 the kernel S α ′ reduces to the field theory kernel S of [2,3,22]. As indicated, we will distinguish between these two momentum kernels by putting a subscript α ′ on the one of string theory. In the next section we will analyze the properties of this kernel.

2.15)
This provide a general form of the closed/open string relation between external gauge bosons and gravitons at tree-level. When restricted to the case of graviton external states the field theory limit of this expression reduces to the form derived in [3,22]. As seen from the above derivation, expression (2.15) is actually independent of the value of j. This j-independence reflects the arbitrariness in the number of contours one closes to the left around the branch point at z = 0 or to the right around the branch point at z = 1. As further explained below, independence under shifts of j is a consequence of the monodromy relations [8] that are satisfied by the color-ordered string amplitudes.

Properties of the momentum kernel
The S α ′ kernel has a number of fundamental properties that correspond to those of the field theory kernel S. In field theory these properties ensure, for instance, the correct cancellation of poles between products of color-ordered gauge amplitudes and correct factorization properties compatible with on-shell recursion techniques. The S α ′ version of these properties can be seen to hold by the same kind of arguments used for the field theory S kernel in [2,3,22], and will therefore not be repeated here.
(1) Reflection symmetry: where p is a massless momentum and σ and γ are arbitrary permutations of the k labels {1, . . . , k}.
(4) The shifting-formula for j: By using that formula (2.15) is independent of j we obtain the following relation, which is valid for any 2 ≤ j ≤ n − 2: . . , i n−2 |β(i j+1 , . . . , i n−2 )] k n−1 × A n (γ(i 2 , . . . , i j ), 1, n − 1, β(i j+1 , . . . , i n−2 ), n) They are particular cases of linear monodromy relations satisfied by the color-ordered amplitudes. Conversely, monodromy relations are necessary and sufficient for proving the j-independence of the general KLT formula (2.15). Such monodromy properties are generated by the contour deformations as discussed in section 2.1. As mentioned, property (1)-(4) are also satisfied in field theory simply by replacing S α ′ by its field theory limit S and replacing the color-ordered string amplitudes A n by their corresponding field theory limits A n . (3.5)

Soft limit of graviton amplitudes at tree-level
In this and in the following section we make some observations on the field theory limit.
Interestingly, tree-level gravity amplitudes have a universal behavior when taking one graviton to be soft, a classic result due to Weinberg [4,29]  For definiteness take leg n to be soft, then the 'soft factor' is given by the sum where S YM (q L , n ± , i) is the corresponding soft factor for Yang-Mills theory lim k ± s →0 A n (· · · , a, s ± , b, · · · ) = S YM (a, s ± , b) A n−1 (· · · , a, b, · · · ) . (4. 3) It depends on the helicity of the soft gluon [30] and is given by where ǫ is the polarization vector of the gluon with momentum k. The covariant expression for the soft factor of the graviton is therefore where the graviton polarization tensor ǫ ±± has been split into a product of Yang-Mills polarizations ǫ ± ⊗ ǫ ± . This expression shows explicitly that the gravity soft factor in (4.2) is independent of the choice of reference momenta q L and q R . Using the soft factor of the graviton changes by which vanishes by momentum conservation and transversally of the polarization vectors. Using standard spinor-helicity notation these soft factors reads (up to normalization constants) . (4.8)

A more crossing-symmetric KLT relation from soft limits
A KLT relation with (n − 2)! 2 terms for the n-graviton amplitude was recently considered in [2]. This expression has a higher degree of manifest crossing symmetry. Ignoring overall normalization constants, it reads  On-shell this expression is of course ill-defined since then s 12...n−1 = k 2 n = 0. However, the numerator also vanishes because of the annihilation property (3.3). In [2] a prescription for taking the on-shell limit such that the formula (4.9) gives the correct n-point gravity amplitude was provided. In this section we will demonstrate that the soft limit of gravity amplitudes imply that the numerator and denominator indeed vanish at the same rate. This gives an alternative understanding of why the limit in (4.9) is finite and corresponds to the proper n-point gravity amplitude. Note that we are using a ≈ to remind ourself that the equality is in terms of a limiting procedure.
A n (1, 2, 3 . . . , n) + A n (1, 3, 4, . . . , n, 2) + · · · + A n (1, n, 2, 3, . . . , n − 1) = 0 . that this kernel defines a set of linear equations that annihilate color-ordered Yang-Mills amplitudes: it is the generator of monodromy relations between these amplitudes. The kernel has also an interesting connection relating to kinematic factors. Although no natural origin of this momentum kernel is known from the structure of the (effective) gravity and gauge theory Lagrangians, we have shown that this object arises naturally when one constructs closed (type II or heterotic) string amplitudes. This parallels (and complements) the S-matrix based motivation for this kernel in [2]. The very simple product form of this kernel makes it easy to implement the KLT-form of the amplitudes in (2.15), the annihilation relations in (3.3), and the definition of the kinematic factors in (5.11).
Our string-based derivation of the annihilation property and, consequently, the monodromy relations between color-ordered amplitudes did not use any detailed properties of the spectrum of the theory. The momentum kernel follows in a universal way from the phases of the operator product expansion. We therefore expect that such a kernel will enter in relations between ordered correlators in other contexts that are based on a conformal field theory, as in solid state physics. It would be interesting to have this possible application elucidated in greater detail.